变分计算例子

解答见原网页 变分计算

1. Find the path that minimizes the arclength of the curve between $ (x_0,y_0) = (0,0)\, $ and $ (x_1,y_1) = (1,1)\, $.
2. Find the extrema of $ x^2+y^2+z^2\, $ subject to the constraint $ x^2+2y^2-z^2-1=0\, $.
3. Find the maximum of $ xy^2z^2\, $ subject to the constraint $ x+y+z=12\, $.
4. Write the Euler-Lagrange equation for $ L(x,y,z,y',z',y'',z'',y''',z''',...,y^{(k)},z^{(k)})\, $.
5. Constraint problem: Minimize $ T(y)=\int_0^1\left(y'^2+x^2\right)\,dx\, $ s.t. $ K(y)=\int_0^1y^2\,dx=2\, $.
6. Derive the Euler-Lagrange equation from the attempt to minimize the functional

   $ T(y)=\int_a^b L(y,y',x)\,dx\, $

7. Minimize the functional from classical mechanics: $ \int_{t_1}^{t_2}(\mathrm{Kinetic\,Energy} - \mathrm{Potential\,Energy})\, $
8. Find the extrema of $ \int_a^b \frac{y'^2}{x^3}\,dx\, $.
   
9. Find the extrema of $ \int_a^b (y^2 +y'^2 + 2y e^x) \,dx\, $.
   
10. Show that the first variation $ \delta J(y_0,h)\, $ satisfies the homogeneity condition $ \delta J(y_0, \alpha h) = \alpha \delta J(y_0, h), \alpha \isin \mathbb{R}\, $. $ J:V\to R'\, $, where $ V\, $ is a normed linear space, is linear if $ J(y_1+y_2) = J(y_1) + J(y_2), y_1,y_2\isin V\, $ and $ J(\alpha y_1) = \alpha J(y_1), \alpha \isin R', y_1\isin V\, $. Which of the following are functionals on $ C^{-1}[a,b]\, $ are linear?

    (a) $ J(y)=\int_a^b y y' dx\, $

    (b) $ J(\alpha y) = \int_a^b (4y'^2 + 2(\alpha y))dx\, $

    (c) $ J(y) = e^{y(a)}\, $

    (d) The set of all continuous functions on $ [0,1]\, $ satisfying $ f(0)=0\, $

    (e) The set of all continuous functions on $ [0,1]\, $ satisfying $ f(1)=1\, $

11. Find the extremal for $ J(y)=\int_1^2 \frac{\sqrt{1+y'^2}}{x} dx, y(1)=0, y(2)=1\, $
12. Compute the first variation of $ J(y)=\int_a^b yy' dx\, $
13. Compute the first variation of $ J(y)=\int_a^b (y'^2+2y)dx\, $
14. Compute the first variation of $ J(y)=e^{y(a)}\, $
15. Minimize $ J(y) = \int_0^\infty (y^2 + y'^2 + (y''+y')^2)dx, y(0)=1, y'(0)=2, y(\infty)=0, y'(\infty)=0\, $
16. Find the extremals of $ J(y) = \int_0^1(yy'+y''^2)dx, y(0)=0, y'(0)=1, y(1)=2, y'(1)=4\, $
17. Find the Euler-Lagrange equation for $ J(y,z)=\int_a^b\left[ y''z' + xyz'' + z'''y^2\right] dx\, $
18. Minimize $ J(y)=\int_0^1(1+y''^2)dx, y(0)=0,y'(0)=1,y(1)=1,y'(1)=1\, $
19. Minimize $ J(y)=\int 2\pi y \sqrt{1+y'^2} dx\, $
20. Obtain a necessary condition for a function $ y\isin C[a,b]\, $ to be a local minimum of the functional

    $ J(y) = \iint\limits_R K(s,t) y(s) y(t) ds dt + \int_a^b y(t)^2dt-2\int_a^b y(t) f(t)dt\, $

21. Find the Euler equation for the functional

    $ J(u)=\iint\limits_G\left[u_x^2+u_y^2+2f(x,y)u(x,y)\right]dxdy\, $

    where $ G\, $ is a closed region in the $ xy\, $ plane and $ u\, $ has continuous second partial derivatives.
    

22. Find the extremal of the functional $ J(y)=\int_0^\pi\left[y'(x)\right]^2dx\, $ subject to the constraint $ \int_0^\pi \left[ y(x)\right]^2dx=1, y(0)=y(\pi)=0\, $.
23. Determine the function $ \hat{y}\isin C^2[0,1]\, $ that minimizes the functional $ J(y)=\int_0^1\left[y'(x)\right]^2dx+[y(1)]^2, y(0)=1, h(0)=0\, $.
24. Let $ J:A\to\mathbb{R}\, $ be a functional on a subset $ A\, $ of a normed linear space $ V\, $.

    (a) Define precisely the first variation $ \delta J(y_0,h)\, $ of $ J\, $ at $ y_0\, $ and admissible $ h(x)\, $.

    (b) Show that if $ \delta J(y_0,h)\, $ exists for a certain admissible $ h\isin V\, $, then $ \delta J(y_0,\alpha h)\, $ also exists for every real number $ \alpha\ $, and $ \delta J(y_0,\alpha h)=\alpha \delta J(y_0,h)\, $.

25. Compute the first variation $ \delta J(y,h)\, $ for $ y\isin C[0,1]\, $: $ J(y)=e^{y(0)}\, $
26. Compute the first variation $ \delta J(y,h)\, $ for $ y\isin C[0,1]\, $: $ J(y)=\int_0^1\int_0^1\sin(xt)y(x)y(t)dxdt\, $
27. Compute the first variation $ J(y) = \int_0^1 (3y^2 + x) dx + y^2(0), y_0(x) = x, h(x)=x+1\, $